Found problems: 85335
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.
Define a sequence $F_n$ such that $F_1 = 1$, $F_2 = x$, $F_{n+1} = xF_n + yF_{n-1}$ where and $x$ and $y$ are positive integers. Suppose $\frac{1}{F_k}= \sum_{n=1}^{\infty}\frac{F_n}{d^n}$ has exactly two solutions $(d, k)$ with $d > 0$ is a positive integer. Find the least possible positive value of $d$.
Let $n> 2$ be a natural. Given $2n$ points in the plane, no $3$ are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle?
[hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?[/hide]
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.
The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that:
$\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\,
\textbf{(B) }x>2\text{ or }x<-2$
$\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\,
\textbf{(D) }x>1\text{ or }x<-2\qquad$
$\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$
Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.
A hinged convex quadrilateral was made of four slats. Then, two points on its opposite sides were connected with another slat, but the structure remained non-rigid. Does it follow from this that this quadrilateral is a parallelogram?
[i]Proposed by A. Zaslavsky[/i]
[center][img width="40"]https://i.ibb.co/dgqSvLQ/Screenshot-2023-03-09-231327.png[/img][/center]
[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions:
(i) $\frac{p+1}{2}$ is a prime number.
(ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer.
[b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.
How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime.
[i]2019 CCA Math Bonanza Lightning Round #1.1[/i]
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $
Find the sum of the greatest common factor and the least common multiple of $12$ and $18$.
Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds $T_{max}$. One end of the cord is attached to a fixed point, the other is attached to an object of mass $3m$. If a second, smaller object of mass m moving at an initial speed $v_0$ strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of $v_f$. All motion occurs on a horizontal, frictionless surface.
Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision.
$ \textbf{(A)}\ 1/4 \qquad\textbf{(B)}\ 1/3 \qquad\textbf{(C)}\ 1/2 \qquad\textbf{(D)}\ 3/4 \qquad\textbf{(E)}\ \text{none of the above} $
The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.
Through a point $P$ within a triangle $ABC$ the lines $l, m$, and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$, $m$ intersects $AC$ in $R$, and $n$ intersects $AB$ in $S$, then the points $Q, R$, and $S$ are collinear.
Show that a graph with 2n points and $n^2 + 1$ edges necessarily contains a 3-cycle, but that we can find a graph with 2n points and $n^2$ edges without a 3-cycle.
please prove it without induction .
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
Let \[x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.\] Find $(x+1)^{48}$.
It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.
[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label("$1$",(1,3));
label("$9$",(2,2));
label("$14$",(3,1));
label("$5$",(4,2));[/asy]
For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and
$$
(s(n))^2=m \text { and }(s(m))^2=n .
$$
Let $ a,b,c $ be positive real numbers, prove the inequality:
$ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $
Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that
(i) If $n = 4$, then
\[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\]
(ii) If $n = 6$, then
\[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]